124 research outputs found
Hierarchical ordering of reticular networks
The structure of hierarchical networks in biological and physical systems has
long been characterized using the Horton-Strahler ordering scheme. The scheme
assigns an integer order to each edge in the network based on the topology of
branching such that the order increases from distal parts of the network (e.g.,
mountain streams or capillaries) to the "root" of the network (e.g., the river
outlet or the aorta). However, Horton-Strahler ordering cannot be applied to
networks with loops because they they create a contradiction in the edge
ordering in terms of which edge precedes another in the hierarchy. Here, we
present a generalization of the Horton-Strahler order to weighted planar
reticular networks, where weights are assumed to correlate with the importance
of network edges, e.g., weights estimated from edge widths may correlate to
flow capacity. Our method assigns hierarchical levels not only to edges of the
network, but also to its loops, and classifies the edges into reticular edges,
which are responsible for loop formation, and tree edges. In addition, we
perform a detailed and rigorous theoretical analysis of the sensitivity of the
hierarchical levels to weight perturbations. We discuss applications of this
generalized Horton-Strahler ordering to the study of leaf venation and other
biological networks.Comment: 9 pages, 5 figures, During preparation of this manuscript the authors
became aware of a related work by Katifori and Magnasco, concurrently
submitted for publicatio
Numerical Approach to Central Limit Theorem for Bifurcation Ratio of Random Binary Tree
A central limit theorem for binary tree is numerically examined. Two types of
central limit theorem for higher-order branches are formulated. A topological
structure of a binary tree is expressed by a binary sequence, and the
Horton-Strahler indices are calculated by using the sequence. By fitting the
Gaussian distribution function to our numerical data, the values of variances
are determined and written in simple forms
Trading inference effort versus size in CNF Knowledge Compilation
Knowledge Compilation (KC) studies compilation of boolean functions f into
some formalism F, which allows to answer all queries of a certain kind in
polynomial time. Due to its relevance for SAT solving, we concentrate on the
query type "clausal entailment" (CE), i.e., whether a clause C follows from f
or not, and we consider subclasses of CNF, i.e., clause-sets F with special
properties. In this report we do not allow auxiliary variables (except of the
Outlook), and thus F needs to be equivalent to f.
We consider the hierarchies UC_k <= WC_k, which were introduced by the
authors in 2012. Each level allows CE queries. The first two levels are
well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in
KC, that is, f is represented by the set of all prime implicates, while UC_1 =
WC_1 is the same as UC, the class of unit-refutation complete clause-sets
introduced by del Val 1994. We show that for each k there are (sequences of)
boolean functions with polysize representations in UC_{k+1}, but with an
exponential lower bound on representations in WC_k. Such a separation was
previously only know for k=0. We also consider PC < UC, the class of
propagation-complete clause-sets. We show that there are (sequences of) boolean
functions with polysize representations in UC, while there is an exponential
lower bound for representations in PC. These separations are steps towards a
general conjecture determining the representation power of the hierarchies PC_k
< UC_k <= WC_k. The strong form of this conjecture also allows auxiliary
variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the
separation results from the discontinued arXiv:1302.442
Hierarchical Analysis of Branching Patterns
制度:新 ; 報告番号:甲3309号 ; 学位の種類:博士(理学) ; 授与年月日:2011/3/15 ; 早大学位記番号:新561
The Horizontal Tunnelability Graph is Dual to Level Set Trees
Time series data, reflecting phenomena like climate patterns and stock prices, offer key insights for prediction and trend analysis. Contemporary research has independently developed disparate geometric approaches to time series analysis. These include tree methods, visibility algorithms, as well as persistence-based barcodes common to topological data analysis. This thesis enhances time series analysis by innovatively combining these perspectives through our concept of horizontal tunnelability. We prove that the level set tree gotten from its Harris Path (a time series), is dual to the time series' horizontal tunnelability graph, itself a subgraph of the more common horizontal visibility graph. This technique extends previous work by relating Merge, Chiral Merge, and Level Set Trees together along with visibility and persistence methodologies. Our method promises significant computational advantages and illuminates the tying threads between previously unconnected work. To facilitate its implementation, we provide accompanying empirical code and discuss its advantages
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